Vanishing Theorems, Support Conditions, and Boundary Problems for \(\overline\partial\) on Weak Z(q) Domains
Bibliographic record
Abstract
Let \( X \) be a complex manifold of complex dimension \( n \geq 2 \), and let \( \Omega \Subset \mathcal{X} \) be a relatively compact domain with smooth boundary that satisfies the weak \( Z(q) \)-condition. Assume \( \mathcal{F} \) is a holomorphic line bundle over \( X \), and denote by \( \mathcal{F}^{\otimes m} \) its \( m \)-th tensor power for some positive integer \( m \). Provided there exists a strongly plurisubharmonic function defined in a neighborhood of the boundary \( b\Omega \), it is possible to obtain solutions to the \( \overline{\partial} \)-equation within \( \Omega \), under support conditions, for \((p,q)\)-forms with \( q \geq 1 \) taking values in \( \mathcal{F}^{\otimes m} \). Additionally, we study the solvability of the boundary \( \overline{\partial}_b \)-problem on weak \( Z(q) \)-domains with smooth boundary in the setting of Kähler manifolds. Moreover, an extension theorem for \( \overline{\partial}_b \)-closed differential forms will be proven.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".